Abstract:The properties of $\Bbb R$-factorizable groups and their subgroups are studied. We show that a locally compact group $G$ is $\Bbb R$-factorizable if and only if $G$ is $\sigma $-compact. It is proved that a subgroup $H$ of an $\Bbb R$-factorizable group $G$ is $\Bbb R$-factorizable if and only if $H$ is $z$-embedded in $G$. Therefore, a subgroup of an $\Bbb R$-factorizable group need not be $\Bbb R$-factorizable, and we present a method for constructing non-$\Bbb R$-factorizable dense subgroups of a special class of $\Bbb R$-factorizable groups. Finally, we construct a closed $G_{\delta }$-subgroup of an $\Bbb R$-factorizable group which is not $\Bbb R$-factorizable.
Keywords: $\Bbb R$-factorizable group, $z$-embedded set, $\aleph _0$-bounded group, $P$-group, Lindel\"of group
AMS Subject Classification: Primary 54H11, 22A05; Secondary 22D05, 54C50